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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On symmetric products of curves


Author: F. Bastianelli
Journal: Trans. Amer. Math. Soc. 364 (2012), 2493-2519
MSC (2010): Primary 14E05, 14Q10; Secondary 14J29, 14H51, 14N05
Published electronically: January 19, 2012
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Abstract: Let $ C$ be a smooth complex projective curve of genus $ g$ and let $ C^{(2)}$ be its second symmetric product. This paper concerns the study of some attempts at extending to $ C^{(2)}$ the notion of gonality. In particular, we prove that the degree of irrationality of $ C^{(2)}$ is at least $ g-1$ when $ C$ is generic and that the minimum gonality of curves through the generic point of $ C^{(2)}$ equals the gonality of $ C$. In order to produce the main results we deal with correspondences on the $ k$-fold symmetric product of $ C$, with some interesting linear subspaces of $ \mathbb{P}^n$ enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of $ C^{(2)}$ when $ C$ is a generic curve of genus $ {6\leq g\leq 8}$.


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Additional Information

F. Bastianelli
Affiliation: Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy
Address at time of publication: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy
Email: francesco.bastianelli@unipv.it, francesco.bastianelli@unimib.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05378-5
PII: S 0002-9947(2012)05378-5
Received by editor(s): February 2, 2010
Received by editor(s) in revised form: April 30, 2010
Published electronically: January 19, 2012
Additional Notes: This work was partially supported by PRIN 2007 “Spazi di moduli e teorie di Lie”, INdAM (GNSAGA), and FAR 2008 (PV) “Varietà algebriche, calcolo algebrico, grafi orientati e topologici”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.